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Evaluation-efficient quantum architecture search with ZX-calculus-based topological reuse

Chenlu Li, Hui Zeng, Dazhi Ding

arXiv:2609.29098v1quant-pheess.SP

TL;DR

Repeated training and evaluation of QAS candidates makes search costly, including when distinct candidates have similar topologies. ZX-QAS combines ternary Gray-code mapping, noise-aware training, and ZX-calculus-based reuse; in molecular tasks, it maintains search quality while reducing real evaluations.

Abstract

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Variational quantum algorithm is a leading approach for quantum chemistry and many-body physics on noisy intermediate-scale quantum devices. The performance depends strongly on the structure of the parameterized quantum circuits. Quantum architecture search (QAS) can automate ansatz design. However, it requires repeated training and evaluation of many candidate circuits, leading to high evaluation cost. In this work, we propose a noise-aware quantum architecture search framework based on ZX-calculus topological reuse (ZX-QAS). The framework encodes the search space with a ternary Gray-code mapping and integrates a noise-aware quantum neural network with a ZX-calculus topological reuse mechanism. The effectiveness of the framework is validated through ground-state energy estimation tasks and one-dimensional transverse-field Ising model tasks under noisy conditions. The results show that the ZX-QAS exhibits stable convergence and remarkably reduces the cost of expensive evaluations under noisy conditions.

I. INTRODUCTION

VQA performance depends on balancing ansatz expressibility, trainability, and hardware overhead, while QAS incurs high costs by repeatedly evaluating candidates. ZX-QAS combines an implicit ternary Gray-code search space with noise-aware training and ZX-calculus-based reuse.

I. INTRODUCTION

  • Ansatz design must balance expressibility, trainability, and hardware overhead because deep, strongly entangling circuits can be noise-sensitive while shallow circuits may lack variational freedom.The passages also identify barren plateaus as a concern for deep circuits.Source: §I · p. 1
  • QAS repeatedly trains and evaluates candidate circuits, making evaluation cost a central bottleneck to scalability and practical utility.Candidates with different gate sequences can share similar topologies and still incur repeated computation.Source: §I · p. 1
  • A practical QAS framework should account for device noise during search and favor ansatzes that remain robust under noisy conditions.Noise can induce qubit flips and decoherence that degrade an algorithm or cause failure.Source: §I · p. 1
  • ZX-QAS combines implicit ternary Gray-code mapping, noise-aware parameter sharing, and ZX-calculus topological reuse to reduce expensive circuit evaluations while maintaining performance.The Gray-code mapping makes one-step mutations local gate changes, while similar candidates can reuse results from evaluated circuits.Source: §I · pp. 1–2

II. RELATED WORK

  • Prior search-space studies compare gate-, layer-, and block-level designs, trading gate-level flexibility against the greater stability of layer- or block-level spaces [23].Practical spaces should represent target correlations while respecting hardware depth and two-qubit-gate constraints.Source: A. Search space · p. 2

A. Search space

  • Search spaces should account for topology, expressibility, trainability, and hardware overhead; existing discrete encodings may lack structural locality and require substantial memory.Circuit connectivity is also identified as a factor in QAS efficiency.Source: A. Search space · p. 2

B. Search and evaluation strategy

QAS efficiency strategies reduce candidate training or use prior information, but repeated evaluation of structurally similar late-stage candidates remains wasteful. ZX-calculus represents circuits as graphs that can be simplified without changing their quantum operation.

B. Search and evaluation strategy

  • QAS efficiency approaches either screen low-potential candidates before full optimization or use prior knowledge to reduce redundant exploration.Examples include proxies, mixtures of experts, generative models, graph-based predictors, and reinforcement learning.Source: B. Search and evaluation strategy · p. 2
  • Late evolutionary search can sample candidates with distinct encodings but similar depths, entanglement configurations, and topological complexities, making repeated evaluations wasteful.Training candidates from scratch to estimate performance is computationally costly.Source: B. Search and evaluation strategy · p. 2

C. ZX-calculus

  • ZX-calculus represents quantum processes as spider-and-edge graphs and simplifies them through rewrite rules without changing the represented operation.The passages name spider fusion, identity removal, color change, and local complementation as examples.Source: C. ZX-calculus · p. 3
  • Earlier ZX-based QAS work focused on circuit simplification or architecture mutation, leaving its use as a structural prior for evaluation-budget management largely unexplored.The paper addresses this gap with implicit ternary Gray-code mapping and topology-aware evaluation reuse.Source: C. ZX-calculus · p. 3

III. METHODOLOGY

The framework targets repeated evaluation of topologically similar ansatzes through implicit search-space mapping and ZX-calculus-based reuse. Its components also include noise-aware architecture training and shared parameters.

III. METHODOLOGY

  • ZX-QAS addresses exponential search-space growth, out-of-memory risk from storing candidates, and repeated evaluation of topologically similar ansatzes.Similar candidates may still require repeated parameter training and gradient evaluation.Source: §III · p. 3
  • The implicit ternary Gray-code mapping stores the current candidate encoding and sparse CNOT lookup table rather than the full set of layer configurations.The method is described as preserving topological locality and stabilizing evolutionary search.Source: §III · p. 3
  • The framework combines noise-aware architecture training and parameter sharing with ZX-calculus-based reuse of evaluations for low-novelty topologies.Its three components are search-space design, a noise-aware quantum neural network, and QAS with topological reuse.Source: §III · p. 3

A. Ternary Gray-code mapping for the search space

The search space pairs single-qubit rotation choices with entangling-gate configurations. Rotation choices use an implicit ternary Gray code, while CNOT configurations come from a lookup table and are decoded on demand.

A. Ternary Gray-code mapping for the search space

  • Each layer combines a single-qubit rotation-gate configuration with an entangling-gate configuration, and gates are decoded only when a candidate is evaluated.The workflow maps rotations with a ternary Gray code and represents entangling gates with a sparse CNOT lookup table.Source: A. Ternary Gray-code mapping for the… · pp. 3, 5, Figure 2 · p. 4
  • Explicitly storing the Cartesian product of rotation and CNOT configurations can cause out-of-memory errors, whereas on-demand decoding avoids that full-space storage.Standard integer encoding can also change several digits at once, causing simultaneous gate changes.Source: A. Ternary Gray-code mapping for the… · pp. 4–5
  • The ternary Gray-code sequence gives adjacent rotation codes a Hamming distance of one, so adjacent indices change only one single-qubit rotation gate.This preserves locality for evolutionary mutations rather than producing abrupt circuit-structure changes.Source: A. Ternary Gray-code mapping for the… · pp. 4–5
  • The integer layer index u maps to a rotation-subspace index and a CNOT-subspace index, enumerating all CNOT configurations before advancing to the next rotation configuration.The rotation entry is selected from the ternary Gray-code sequence and the entangling entry from the CNOT lookup table.Source: A. Ternary Gray-code mapping for the… · p. 5

B. Noise-aware quantum neural networks

The noise-aware quantum neural network pretrains shared-parameter supernet experts under a noise model, then uses them to initialize candidate evaluation and optimization.

B. Noise-aware quantum neural networks

  • SPOS pretrains a noise-aware supernet whose shared parameters initialize candidate circuits for evaluation.The supernet uses K experts with independent parameter pools.Source: B. Noise-aware quantum neural networks · pp. 5–6
  • The modeled hardware noise includes post-gate bit flips, depolarizing errors, and thermal relaxation during gate operations.For CNOT gates, the depolarizing model spans the two-qubit Pauli-error space.Source: B. Noise-aware quantum neural networks · p. 5
  • For each sampled architecture, expert k produces a noisy output state ρ(A, θk) and its energy expectation E_k.The energy is computed as Tr[ρ(A, θk)H].Source: B. Noise-aware quantum neural networks · p. 6
  • Optimization selects the expert with the lowest current energy, then uses ε-greedy exploration and gradient descent before passing pretrained parameters to evolutionary search.The best expert is exploited with probability 1−ε; another is selected with probability ε.Source: B. Noise-aware quantum neural networks · p. 6

C. Quantum architecture search with ZX-calculus-based topological reuse

The search combines variable-depth evolutionary optimization with ZX-graph matching and similarity-based reuse to reduce redundant candidate evaluations.

C. Quantum architecture search with ZX-calculus-based topological reuse

  • Variable-depth NSGA-II search evaluates candidate fitness with ZX-based topological reuse and returns the best architecture found.The scalar fitness ranks candidates during the evolutionary search.Source: C. Quantum architecture search with… · pp. 6–7
  • Variable-length integer chromosomes encode circuit layers, while ternary Gray-code mutations usually map local encoding changes to local gate changes.Tournament selection, crossover, mutation, and elitist population updates generate successive populations.Source: C. Quantum architecture search with… · pp. 6–7
  • After ZX reduction, matching structural signatures identify candidates with the same reduced pattern and allow cached fitness reuse.The signature records graph nodes, types, phases, qubit or row indices, connections, and edge types.Source: C. Quantum architecture search with… · p. 6
  • When signatures do not match, a weighted distance over ZX structural features identifies low-novelty candidates without assuming exact circuit equivalence.The distance weights reduced spider counts and entangling-edge distributions relatively heavily, with depth weighted intermediately.Source: C. Quantum architecture search with… · p. 6
  • For low-novelty candidates, reuse probability rises as search progresses while a minimum real-evaluation probability remains; reused candidates receive a distance- and progress-dependent redundancy penalty.Candidates not identified as low-novelty, or not selected for reuse, undergo normal evaluation.Source: C. Quantum architecture search with… · pp. 6–7
  • The fitness combines energy error, residual ZX spider count, and circuit depth, with candidate parameters initialized from the best supernet pool and fine-tuned using L-BFGS-B.The depth term approximately accounts for execution time, gate-noise accumulation, and decoherence risk.Source: C. Quantum architecture search with… · p. 7

IV. EVALUATION

The evaluation tests ZX-QAS on ground-state energy estimation and the 1D transverse-field Ising model under noisy conditions.

IV. EVALUATION

  • ZX-QAS is evaluated on ground-state energy estimation and one-dimensional transverse-field Ising model tasks.The evaluation is conducted under noisy conditions.Source: §IV · p. 8
  • The noisy-condition comparisons examine error convergence, the relation between ZX topological complexity and energy error, and candidate-ansatz reuse rates.Source: §IV · p. 8
  • Ablation experiments test the effectiveness of the ternary Gray-code implicit mapping mechanism.Source: §IV · p. 8

A. Experimental Setup

The experimental setup uses a baseline noise configuration intended to represent average performance metrics of current mainstream superconducting processors.

A. Experimental Setup

  • The baseline hardware noise configuration is aligned with average performance metrics of current mainstream superconducting quantum processors.Source: A. Experimental Setup · p. 8
  • With λ = 1.0, the setup uses p1 = 0.001, p2 = 0.004, pBF = 0.05, T1 = 100,000 ns, T2 = 50,000 ns, Tg1 = 30 ns, and Tg2 = 80 ns.Source: A. Experimental Setup · p. 8

B. Ground-state energy estimation

Across noisy ground-state energy tasks for H2, LiH, and H4, ZX-QAS reaches errors comparable to or lower than NA-QAS with fewer evaluation steps, although noisy H4 has a slightly higher final error. Its reuse rates show that ZX similarity reuse accounts for much of the evaluation-cost reduction, especially in the larger search spaces.

B. Ground-state energy estimation

  • ZX-QAS preserves a cost–accuracy tradeoff: it reduces repeated evaluations while maintaining stable convergence, with a small noisy-condition accuracy loss reported for H4.The reported H4 difference may relate to noise accumulation and uncertainty in structural-similarity estimation.Source: B. Ground-state energy estimation · pp. 8, 10
  • ZX-QAS reaches comparable or lower energy errors than NA-QAS in most cases with fewer evaluations, but noisy H4 ends slightly higher at 0.082 versus 0.078 hartree.Under noiseless conditions, ZX-QAS slightly outperforms NA-QAS on H4, reaching about 0.044 versus 0.046 hartree.Source: B. Ground-state energy estimation · p. 8, Figure 3 · p. 9
  • Across H2, LiH, and H4, ZX-QAS candidate errors are more concentrated in lower-error regions, with fewer high-error outliers than Original QAS.For H2, ZX-QAS candidates mainly fall between 0.03 and 0.06 hartree; for LiH, they concentrate near 0.01 hartree.Source: B. Ground-state energy estimation · pp. 8–9, Figure 4 · p. 9
  • Energy error is not monotonic in ZX spider count, so spider count serves as a structural regularizer alongside similarity features rather than a standalone quality criterion.Fewer spiders can mean less noise accumulation but insufficient expressibility; more spiders can increase expressibility while adding noise and optimization difficulty.Source: B. Ground-state energy estimation · pp. 9–10
  • Reuse grows as search proceeds: H2 reaches one evaluation at generation 18, LiH declines to about 3–5 per generation, and H4 later reaches about 1–5.H4 has a renewed evaluation increase around generations 8–13, so reuse does not rise monotonically in that larger search space.Source: B. Ground-state energy estimation · p. 10, Figure 5 · p. 11
  • H2 has 40% ZX-similarity reuse and an 11% exact-cache hit rate; H4 and LiH have 48.25% and 47.50% ZX-similarity reuse, respectively.Table I reports 400 EA samples; H2's combined reuse ratio is 51%, while exact-cache hit rates are only 1.25% for H4 and 1.75% for LiH.Source: B. Ground-state energy estimation · p. 10, Table I · p. 11

C. One-dimensional transverse-field Ising model

In the N = 8 1D TFIM benchmark near the quantum critical region, ZX-QAS converges to lower error than NA-QAS under noisy conditions and maintains stable search behavior with and without noise. ZX structural reuse reduces repeated evaluations, while candidate-error distributions suggest spider count alone is not a sufficient performance screen.

C. One-dimensional transverse-field Ising model

D. Ablation analysis of the Gray-code implicit mapping

The ternary Gray-code mapping makes adjacent encodings correspond to local gate changes, while on-demand decoding avoids storing the full Cartesian search space. The ablations report both more local mutations and a substantial memory advantage over explicit storage.

D. Ablation analysis of the Gray-code implicit mapping

  • DH = 1 for every adjacent ternary Gray-code change, versus about 66.68% single-gate changes under standard ternary mapping.In the N = 8 ablation over 38 single-qubit rotation-gate combinations, standard ternary mapping also produces two-gate changes about 22.23% of the time and three-gate changes about 7.41% of the time.Source: D. Ablation analysis of the Gray-code… · p. 13
  • At N = 14, explicit Cartesian storage requires about 367 GB, while the on-demand mapping's memory-saving ratio exceeds 10^8.The inset compares memory saving as qubit count increases.Source: Figure 10 · p. 13, D. Ablation analysis of the Gray-code… · p. 13
  • The mapping preserves the search space's expressibility while making mutation or local search more likely to produce local circuit changes.The paper links this local continuity with preserving locally useful structures from existing candidates.Source: D. Ablation analysis of the Gray-code… · pp. 13–14
  • On-demand decoding generates rotation and entangling gates from an integer index only when a candidate is evaluated, avoiding storage of the full Cartesian space.It stores the current candidate encoding and CNOT lookup table rather than a complete library of candidate circuits.Source: D. Ablation analysis of the Gray-code… · p. 14

V. CONCLUSION

ZX-QAS combines an implicit ternary Gray-code search space with noise-aware evaluation and probabilistic ZX topological reuse. The reported experiments show stable convergence across molecular energy estimation and 1D TFIM tasks, with evaluation-cost savings above 40%.

V. CONCLUSION

  • ZX-QAS combines ternary Gray-code implicit mapping, a noise-aware quantum neural network, and probabilistic reuse based on ZX structural similarity.Source: §V · p. 14
  • ZX-QAS maintains stable convergence in molecular ground-state estimation and 1D TFIM, with evaluation-cost savings above 40%.Source: §V · p. 14
  • Probabilistic topological reuse may introduce approximation errors, so its similarity threshold, reuse penalty, and minimum real-evaluation probability should be adjusted to task scale.Source: §V · p. 14
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